C • STATISTICAL THEORIES OF TURBULENCE 



particular, it is found that the correlation function is 



m,yi,^,t) =j^^^ III Ria,h,c,0) 



exp 



(^ - ar + iv- b'-) + (r - cv 



Svt 



dadbdc (15-4) 



This last formula can be used to evaluate the asymptotic behavior of the 

 correlation function for large values of t. 



A simpler approach to the problem is to use the spectral tensor. In 

 fact, Eq. 15-2 shows that the pressure terms Pik must be dropped when 

 the nonlinear effect represented by Tik is negligible in the spectral equa- 

 tion (Eq. 11-6), which becomes simply 



^ = -2pK'Fij (15-5) 



The general solution of this equation is 



Fij{K^, t) = FijiKrr,, ^o)^-^-^^'-'") (15-6) 



From this, we may calculate the correlation tensor by a Fourier trans- 

 formation. For large values of t — to, only small values of k are important. 

 Thus one may try to expand Fik in powers of Km and retain only the 

 lowest terms. 



Following this method, Batchelor and Proudman [31] found that the 

 longitudinal correlation coefficient /(r, t) is of the form (Eq. 14-13) for 

 isotropic turbulence and certain very special cases of anisotropic turbu- 

 lence. Previous to this investigation, Batchelor and Townsend [4-3] com- 

 pared the experimental curve for f(r, t) with the Gaussian curve (Eq. 

 14-13) and found good agreement. At that time, this agreement was ex- 

 plained by assuming Fij{Km, t) to be essentially expandible as a Taylor 

 series in k™. Since this assumption is now found to be not true in general, 

 other tentative explanations are suggested by Batchelor and Proudman 

 [31]. A critical examination of this problem is clearly warranted. 



Early period of decay. Much experimental information is available 

 during the early part of the decay process. Recently, Stewart and Town- 

 send [22] summarized their results and compared them with some of the 

 above self-preserving hypotheses. They cautioned against the assump- 

 tion of complete self-preservation, but did not include case (c) in their 

 discussion, which seems to fit all their experimental findings. 



In Fig. C,15b, the law of decay observed by Stewart and Townsend 

 [22] is presented. Although the variation of X^ and yr^ both follow the 

 linear law, as they would in the case of complete similarity, the origin 

 of time (or x axis) must he taken differently for the two straight fines. It 



< 232 > 



